Grade 6
  1. Number System  1.1. Whole Numbers  1.1.1. Place Value and Face Value  1.1.2. Comparing and Ordering Numbers  1.1.3. Rounding Numbers  1.1.4. Properties of Whole Numbers (Associative, Commutative, Distributive)  1.2. Integers  1.2.1. Introduction to Integers  1.2.2. Addition and Subtraction of Integers  1.2.3. Multiplication and Division of Integers  1.2.4. Properties of Integer Operations  1.3. Fractions  1.3.1. Types of Fractions  1.3.2. Equivalent Fractions  1.3.3. Simplification of Fractions  1.3.4. Addition and Subtraction of Fractions  1.3.5. Multiplication and Division of Fractions  1.3.6. Comparing and Ordering Fractions  1.4. Decimals  1.4.1. Understanding Decimals  1.4.2. Conversion between Fractions and Decimals  1.4.3. Addition and Subtraction of Decimals  1.4.4. Multiplication and Division of Decimals  1.4.5. Comparing and Ordering Decimals
  2. Algebra  2.1. Basics of Algebra  2.1.1. Variables and Constants  2.1.2. Expressions and Equations  2.1.3. Algebraic Terms and Coefficients  2.2. Solving Simple Equations  2.2.1. Balancing Equations  2.2.2. Solving One-Step Equations  2.2.3. Solving Two-Step Equations  2.3. Patterns and Sequences  2.3.1. Identifying Patterns  2.3.2. Arithmetic Sequences
  3. Ratio and Proportion  3.1. Ratios  3.1.1. Understanding Ratios  3.1.2. Equivalent Ratios  3.1.3. Simplifying Ratios  3.2. Proportion  3.2.1. Introduction to Proportion  3.2.2. Direct and Inverse Proportions  3.3. Percentages  3.3.1. Converting Ratios to Percentages  3.3.2. Percentage Increase and Decrease  3.3.3. Discount and Profit Calculations  3.3.4. Simple Interest Calculations in Percentages in Ratio and Proportion
  4. Geometry  4.1. Basic Geometric Shapes  4.1.1. Points, Lines, and Angles  4.1.2. Types of Angles  4.1.3. Intersecting and Parallel Lines  4.2. Triangles  4.2.1. Types of Triangles  4.2.2. Properties of Triangles  4.2.3. Sum of Angles in a Triangle  4.2.4. Congruence of Triangles  4.3. Quadrilaterals  4.3.1. Types of Quadrilaterals  4.3.2. Properties of Quadrilaterals  4.4. Circles  4.4.1. Radius and Diameter  4.4.2. Circumference and Area  4.4.3. Arcs and Chords  4.4.4. Sector and Segment
  5. Mensuration  5.1. Perimeter  5.1.1. Perimeter of Simple Shapes  5.2. Area  5.2.1. Area of Rectangles and Squares  5.2.2. Area of Triangles and Parallelograms  5.2.3. Area of Composite Figures  5.2.4. Area of Trapezium  5.3. Volume  5.3.1. Volume of Cubes and Cuboids  5.3.2. Volume of Cylinders  5.3.3. Volume of Cones  5.3.4. Surface Area of Simple Solids
  6. Data Handling  6.1. Introduction to Data  6.1.1. Types of Data  6.1.2. Organizing Data  6.1.3. Frequency Distribution  6.2. Graphs and Charts  6.2.1. Bar Graphs  6.2.2. Line Graphs  6.2.3. Pictographs  6.2.4. Pie Charts  6.3. Mean, Median, and Mode  6.3.1. Calculating Mean  6.3.2. Finding Median  6.3.3. Identifying Mode  6.3.4. Range of Data in Mean, Median, and Mode
  7. Probability  7.1. Basics of Probability  7.1.1. Understanding Probability  7.1.2. Probability of Simple Events  7.1.3. Experimental and Theoretical Probability
  8. Practical Geometry  8.1. Construction of Shapes  8.1.1. Constructing Triangles  8.1.2. Constructing Quadrilaterals  8.1.3. Constructing Circles and Arcs  8.2. Symmetry  8.2.1. Line of Symmetry  8.2.2. Rotational Symmetry  8.2.3. Translational Symmetry

Grade 6 → Geometry ↓


Circles


Welcome to the world of circles! In our geometry journey, understanding circles is an important milestone. A circle is not just a simple round figure; it is a powerful shape with unique and interesting properties. Let's dive deep into the fascinating world of circles and explore their features, terms, and properties with lots of examples.

What is a circle?

A circle is a perfectly round shape. It is the set of all points in a plane that are at a fixed distance from a given point. This fixed distance is called the radius, and the point is called the center of the circle.

R Center

Important terms related to circles

  1. Centre: The point from which all points on a circle are equidistant. Usually denoted as C
  2. Radius (r): The distance from the center of a circle to any point on its edge.
  3. Diameter (d): A straight line passing through the center of a circle, and whose endpoints lie on the circle. The diameter is twice the length of the radius. 
    d = 2r
  4. Circumference: The distance or length around a circle. 
    c = 2πr
  5. Chord: A line segment whose endpoints are on a circle. A diameter is a special kind of chord. Wire
  6. Arc: A portion of the circumference of a circle. It is described by two points on the circle and the path between them.
  7. Sector: The area bounded by two radii and their intercepted arcs. Imagine it like a slice of pie. Area
  8. Segment: The region enclosed by a chord and an arc between the endpoints of the chord.

Circumference of a circle

One of the most exciting parts about circles is calculating their circumference. The circumference is the perimeter of the circle, which represents the distance around the circle. We use a formula involving the constant π (pi), which is a special number equal to approximately 3.14159.

c = 2πr

Let us understand this with an example:

If the radius of a circle is 7 cm, then the circumference is: 

c = 2 × π × 7
  = 2 × 3.14159 × 7
  ≈ 43.98 cm

Area of a circle

Just like we calculate the area for other shapes, we can determine the area of a circle as well. The area of a circle tells us the size of the area enclosed by the circle. The formula for area is:

a = πr²

Example:

If the radius of a circle is 5 cm, then its area is: 

a = π × (5)²
  = 3.14159 × 25
  ≈ 78.54 cm²

Understanding pi (π)

π is a fascinating and important number in mathematics. It is an irrational number, meaning that it has an infinite number of decimal places without repeating. Typically, π is approximated as 3.14 or the fraction 22/7.

Observing the properties of a circle

Imagine you're cutting a slice of pizza; each slice represents a sector. Now, if you run your finger across the crust of that slice, you're creating an arc. If you draw a line from one piece of crust to the other without going through the middle, that's a chord.

Here's an interactive way to understand it:

Arch Area Wire

Tangent to a circle

A tangent is a straight line that touches the circle at exactly one point. This point is known as the point of tangency. The tangent line is always perpendicular to the radius at the point of tangency.

tangent line radius

Properties of circles

  • All radii of a circle are equal. If you measure the distance from the center to the edge in different directions, it will always be the same.
  • The longest chord of a circle is its diameter.
  • The circumference is directly proportional to the radius.
  • In a circle, the number of symmetry lines passing through the centre is infinite.

Using circle knowledge

Understanding circles isn't just for solving math problems; it has real-life applications, too. For example, engineers use circles when designing wheels and gears. Architects incorporate circular structures like domes and arches into their designs.

Example: Circle real-world problem

Let's consider a real-world problem. Suppose you are given the task of building a circular track field with a specific area, say 706.5 square meters. You have to find the required radius.

We know the formula for area: 

a = πr²

given: 

A = 706.5
π ≈ 3.14159

Substitute the values to find r: 

706.5 = πr²
R² = 706.5 / π
R² ≈ 225
r ≈ 15

Therefore, the radius should be around 15 m to get the desired area.

Conclusion

Circles are an integral part of geometry. They are simple yet incredibly versatile and important in mathematics and beyond. By understanding the properties, formulas and applications of circles, we can better understand their role in the real world, whether designing, building or solving mathematical puzzles, making accurate calculations. Happy exploring the world of circles!


Grade 6 → 4.4


U
username
0%
completed in Grade 6

← Prev (4.3.2)
Properties of Quadrilaterals
Next (4.4.1) →
Radius and Diameter

Comments 



Circles

https://www.buddymath.com/g6/4.4?bmal=en